Lesson TRIGONOMETRIC FUNCTIONS OF ANGLES GREATER THAN 90 DEGREES
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This lesson provides an overview of trigonometric functions for angles greater than 90 degrees. REFERENCES <a href = "http://oakroadsystems.com/twt/refangle.htm" target = "_blank">http://oakroadsystems.com/twt/refangle.htm</a> <a href = "http://library.thinkquest.org/20991/alg2/trig.html" target = "_blank">http://library.thinkquest.org/20991/alg2/trig.html</a> <a href = "http://www.analyzemath.com/Angle/reference_angle.html" target = "_blank">http://www.analyzemath.com/Angle/reference_angle.html</a> <a href = "http://www.intmath.com/Trigonometric-functions/6_Trigonometry-functions-any-angle.php" target = "_blank">http://www.intmath.com/Trigonometric-functions/6_Trigonometry-functions-any-angle.php</a> <a href = "" target = "_blank"></a> <a href = "" target = "_blank"></a> TRIGONOMETRIC FUNCTIONS OF A RIGHT TRIANGLE Sine of an angle = opposite / hypotenuse. Cosine of an angle = adjacent / hypotenuse. Tangent of an angle = opposite / adjacent. RECIPROCAL TRIGONOMETRIC FUNCTIONS OF A RIGHT TRIANGLE Cosecent of an angle = 1/Sine of the angle = hypotenuse / opposite. Secant of an angle = 1/Cosine of the angle = hypotenuse / adjacent. Cotangent of an angle = 1/Tangent of the angle = adjacent / opposite. <a href = "http://theo.x10hosting.com/examples/Trigonometry/RightTriangle2.html" target = "_blank">Picture of Basic and Reciprocal Trigonometric Functions</a> QUADRANTS A circle is divided into 4 quadrants. Angles are assigned to quadrants as follows: Quadrant 1 contains positive angles between 0 and 90 degrees. Quadrant 2 contains positive angles between 90 and 180 degrees. Quadrant 3 contains positive angles between 180 and 270 degrees. Quadrant 4 contains positive angles between 270 and 360 degrees. Quadrant 1 contains negative angles between -270 and -360 degrees. Quadrant 2 contains negative angles between -180 and -270 degrees. Quadrant 3 contains negative angles between -90 and -180 degrees. Quadrant 4 contains negative angles between 0 and -90 degrees. 0 degrees is in the same position as 360 degrees and -360 degrees. <a href = "http://theo.x10hosting.com/examples/Trigonometry/Quadrants1.html" target = "_blank">Picture of Quadrants</a> Angles on the borders can be assigned to either quadrant, but are special cases because either the adjacent side or the opposite side of the angle will be 0. Opposite Side equals 0 if angle = 0 or 180 degrees. Adjacent Side equals 0 if angle = 90 or 270 degrees. Opposite Side of the angle is always in the vertical direction either up or down from the x-axis. Adjacent Side of the angle is always in the horizontal direction either left or right of the y-axis. <a href = "http://theo.x10hosting.com/examples/Trigonometry/Borders1.html" target = "_blank">Picture of Angle at the border between Quadrant 4 and Quadrant 1 (0 degrees / 360 degrees)</a> <a href = "http://theo.x10hosting.com/examples/Trigonometry/Borders2.html" target = "_blank">Picture of Angle at the border between Quadrant 1 and Quadrant 2 (90 degrees / -270 degrees)</a> <a href = "http://theo.x10hosting.com/examples/Trigonometry/Borders3.html" target = "_blank">Picture of Angle at the border between Quadrant 2 and Quadrant 3 (180 degrees / -180 degrees)</a> <a href = "http://theo.x10hosting.com/examples/Trigonometry/Borders4.html" target = "_blank">Picture of Angle at the border between Quadrant 3 and Quadrant 4 (270 degrees / -90 degrees)</a> SIGNS OF TRIGONOMETRIC FUNCTIONS The Sign of the Trigonometric function depends on the quadrant that the angle is in. The sign is dependent on whether the opposite side or the adjacent side of the right triangle formed is positive or negative. The sign of the Sine and Cosecant function is determined by the sign of the opposite side. The sign of the Cosine and Secant function is determined by the sign of the adjacent side. The sign of the Tangent and Cotangent function is determined by the signs of the opposite and adjacent side. If they are different, then the Tangent function is negative. If they are the same, then the Tangent function is positive. The hypotenuse is always positive. The signs in the table below are for angles that are within each quadrant border. They do not apply to the signs of the angles at the borders. Those are special as indicated earlier. A table of signs is shown below: <pre> Function Quadrant 1 Quadrant 2 Quadrant 3 Quadrant 4 Sine and Cosecant + + - - Cosine and Secant + - - + Tangent and Cotangent + - + - </pre> <a href = "http://theo.x10hosting.com/examples/Trigonometry/SignsOfTrigFunctionsInEachQuadrant.html" target = "_blank">Picture of Signs of Trigonometric Functions in each Quadrant</a> POSITIVE AND NEGATIVE ANGLES Positive Angles start from 0 degrees and rotate counterclockwise. Negative Angles start from 0 degrees and rotate clockwise. You can convert your negative angle to its equivalent positive angle by adding 360 degrees to it until it turns positive. Once it is positive, you can treat it the same as you would any other positive angle in the quadrant that it is in. Example: Angle is -135 degrees. Add 360 degrees to it until it turns positive. It turns positive the first time we add 360 degrees to it. The equivalent positive angle is 225 degrees. It is in quadrant 3. The trigonometric functions of the angle are treated exactly the same way you would treat the trigonometric functions of any other positive angle in quadrant 3. The reference angle is the same. The trigonometric functions are the same. Sin(-135) = Sin(225) Cos(-135) = Cos(225) Tan(-135) = Tan(225) The reciprocal functions of Cosecant, Secant, and Cotangent are also the same. Cosecant = 1 / Sine Secant = 1 / Cosine Cotangent = 1 / Tangent <a href = "http://theo.x10hosting.com/examples/Trigonometry/PositiveAndNegativeAngleInQuadrant3.html" target = "_blank">Picture of Positive and Negative Angle in Quadrant 3</a> REFERENCE ANGLES Reference Angles are angles that are less than 90 degrees that provide the same trigonometric functions as the original angle except for the signs. The reference angle for an angle in quadrant 1 is equal to the angle. No adjustment is necessary. <a href = "http://theo.x10hosting.com/examples/Trigonometry/ReferenceAngleQuadrant1.html" target = "_blank">Picture of Reference Angle in Quadrant 1</a> The reference angle for an angle in quadrant 2 is equal to 180 degrees minus the angle. <a href = "http://theo.x10hosting.com/examples/Trigonometry/ReferenceAngleQuadrant2.html" target = "_blank">Picture of Reference Angle in Quadrant 2</a> The reference angle for an angle in quadrant 3 is equal to the angle minus 180 degrees. <a href = "http://theo.x10hosting.com/examples/Trigonometry/ReferenceAngleQuadrant3.html" target = "_blank">Picture of Reference Angle in Quadrant 3</a> The reference angle for an angle in quadrant 4 is equal to 360 degrees minus the angle. <a href = "http://theo.x10hosting.com/examples/Trigonometry/ReferenceAngleQuadrant4.html" target = "_blank">Picture of Reference Angle in Quadrant 4</a> PROCEDURE FOR FINDING A REFERENCE ANGLE IF THE ANGLE IS POSITIVE If the angle is greater than 360 degrees, you subtract 360 degrees from it until the angle is less than 360 degrees. If the angle is in the first quadrant, the reference angle is the angle. If the angle is in the second quadrant, the reference angle is 180 degrees minus the angle. If the angle is in the third quadrant, the reference angle is the angle minus 180 degrees. If the angle is in the fourth quadrant, the reference angle is 360 minus the angle. The signs of the trigonometric functions are determined by the quadrant that the angle is in. Always take that into consideration when working with the reference angle. Angle is in Quadrant 1: All Trigonometric functions are positive. Angle is in Quadrant 2: Sine is positive, Cosine is negative, Tangent is negative. Angle is in Quadrant 3: Sine is negative, Cosine is negative, Tangent is positive. Angle is in Quadrant 4: Sine is negative, Cosine is positive, Tangent is negative. PROCEDURE FOR FINDING A REFERENCE ANGLE IF THE ANGLE IS NEGATIVE Add 360 degrees to the angle until the angle becomes positive. Follow the rules for finding the reference angle when the angle is positive. SHORTCUT FOR VERY LARGE POSITIVE ANGLE Divide the angle by 360. Take the integer part of the result and multiply 360 by that. Subtract the result from the angle. Example: Your angle is 15065. Divide by 360 to get 41.8 Multiply 360 by 41 to get 14760 Subtract 14760 from 15065 to get 305. 305 is the angle you need to work with to get your reference angle from. It is in the fourth quadrant. SHORTCUT FOR VERY LARGE NEGATIVE ANGLE Divide the absolute value of the angle by 360. take the integer part of the result and add 1 to it. Multiply 360 by that. Subtract the absolute value of the angle from the result. Example: Your angle is -14815 Divide 14815 by 360 to get 41.15 Add 1 to it to get 42 Multiply 360 by 42 to get 15120 Subtract 14815 from 15120 to get 305. 305 is the angle you need to work with to get your reference angle from. It is in the fourth quadrant. Note that the positive angle of 15065 degrees that I chose is the equivalent of the negative angle of -14815 that I chose. I did that on purpose to show you that the angles are equivalent. Both these angles reduce to the same angle that you need to work with in order to find the reference angle. Both these angles point to the same reference angle (55 degrees). Note that a positive angle of 305 degrees is equivalent to a negative angle of -55 degrees. Just add 360 to the negative angle of -55 degrees and it becomes a positive angle of 305 degrees. <a href = "http://theo.x10hosting.com/examples/Trigonometry/Reduction1.html" target = "_blank">Picture of angle you need to work with for positive angle of 15065 degrees and it's equivalent negative angle of -14815 degrees</a> GRAPH OF TRIGONOMETRIC FUNCTIONS The graph of a trigonometric function shows you the relationship between the angle and the trigonometric function for that angle. The x-axis plots the value of angle. This can be in degrees, or it can be in radians. The y-axis plots the value of the trigonometric function. This is in the form of a ratio of the length of the sides involved in the trigonometric function. This ratio is positive or negative depending on the signs of the adjacent and opposite sides. The signs of the adjacent and opposite sides are dependent on the quadrant that the angle is in. In the following graphs, the values on the x-axis are in radians. The equivalent degrees are shown at the bottom of the vertical lines that intersect with the x-axis every 90 degrees. Degrees are equal to radians * 180 / pi. Radians are equal to degrees * pi / 180. Example: To convert 360 degrees to radians, you would multiply 360 * (pi/180) to get 6.283185307 radians. To convert 6.283185307 radians to degrees, you would multiply 6.283185307 * (180/pi) to get 360 degrees. pi is equal to the constant number of 3.141592654 that defines the relationship between the length of an arc of a circle and its radius. The greek symbol for pi is {{{pi}}}. The translations are as follows: <pre> x value degrees +/- 1.57 +/- 90 +/- 3.14 +/- 180 +/- 4.71 +/- 270 +/- 6.28 +/- 360 +/- 7.85 +/- 450 +/- 9.42 +/- 540 +/- 10.99 +/- 630 +/- 12.56 +/- 720 </pre> Angles and Trigonometric functions of angles are cyclical in nature. The angle repeats its exact same position relative to the 0 degree mark every 360 degrees. The Trigonometric Functions also repeat in a cyclical fashion. Since the basis for the trigonometric functions is the right triangle, they repeat every 90 degrees with the exception of their signs. Every 360 degrees, the trigonometric functions are exactly the same, including the signs. The graph of the trigonometric functions shows this clearly if you allow the x-axis to extend through more than one complete cycle in a positive or negative direction. The following graphs have done that. You will see the same trigonometric function pattern of values every 360 degrees. The base will be from 0 to 360 degrees. That shows one revolution of the angle from 0 to 360 degrees with the associated trigonometric functions for that angle. Look at 360 to 720 degrees and you will see that the same pattern has repeated. Look at -360 degrees to 0 degrees and you will see that the same pattern has repeated. The graphs are shown below: <a href = "http://theo.x10hosting.com/examples/Trigonometry/GraphOfSineFunction.html" target = "_blank">GRAPH OF SINE FUNCTION</a> <a href = "http://theo.x10hosting.com/examples/Trigonometry/GraphOfCosineFunction.html" target = "_blank">GRAPH OF COSINE FUNCTION</a> <a href = "http://theo.x10hosting.com/examples/Trigonometry/GraphOfTangentFunction.html" target = "_blank">GRAPH OF TANGENT FUNCTION</a> <a href = "http://theo.x10hosting.com/examples/Trigonometry/GraphOfCosecantFunction.html" target = "_blank">GRAPH OF COSECANT FUNCTION</a> <a href = "http://theo.x10hosting.com/examples/Trigonometry/GraphOfSecantFunction.html" target = "_blank">GRAPH OF SECANT FUNCTION</a> <a href = "http://theo.x10hosting.com/examples/Trigonometry/GraphOfCotangentFunction.html" target = "_blank">GRAPH OF COTANGENT FUNCTION</a> CALCULATOR Most Scientific Calculators will allow you to find the trigonometric function of any angle by simply entering the angle in degrees or radians, depending on how the calculator is set up. They provide Tangent Function, Sine Function, and Cosine Function at least. Not all provide Cosecant, Secant, or Cotangent Function directly. However, ... The Cosecant Function is equal to 1 divided by the Sine Function. The Secant Function is equal to 1 divided by the Cosine Function. The Cotangent Function is equal to 1 divided by the Tangent Function. You should always check your work through the use of the calculator to see if you did it right. PROCEDURE ON MY TI-30XA SCIENTIFIC CALCULATOR If I want to get the Sine Function of 15065 degrees, I would do the following: Press the DRG key repeatedly until DEG shows on the display. This means I am entering in degrees. Enter 15065. Press the SIN key. The result of -.819152044 is displayed. This is the Sine Function of 15065 degrees. If I want to get the Cosecant Function of 15065 degrees, I would do the following: Display the Sine Function of 15065 degrees as before. Press the DIVIDE key. Enter 1. Press the 2nd Function key. Press the {{{pi}}} key. The little gold letters on top of the {{{pi}}} key are the 2nd function for the {{{pi}}} key which show up as x <---> y which means that I am interchanging -.819152044 with 1 so that instead of dividing -.819152044 by 1, I will be dividing 1 by -.819152044. Press the EQUAL key. The result of -1.220774589 is displayed. This is the Cosecant Function of 15065 degrees. Alternatively, if I want to get the Cosecant Function of 15065 degrees, I would do the following: Display the Sine Function of 15065 degrees as before. Press the STO key. Enter 1. This stores the display of -.819152044 into memory location number 1. Enter 1. Press the DIVIDE key. Press RCL Enter 1. This recalls the number -.819152044 and enters it into the display. Press the EQUAL key. -1.220774589 is displayed. This is the Cosecant Function of 15065 degrees. Your scientific calculator should have equivalent functions built in to it. If you do not have a calculator that has these functions built in, then the <a href = "http://www.1728.com/trigcalc.htm" target = "_blank">ONLINE TRIGONOMETRIC FUNCTION CALCULATOR</a> will be able to help you.
